What truly are length, area and volume? And considerations about divergence in normed spaces

All the "(?)" are parts when i'm not sure at all if what i'm saying is right or not, it's just my intuition.

Part 1

In $\mathbb{R}$, we can define the length of a segment.

In $\mathbb{R}^2$, the 'equivalent' would be (?) the area of a closed shape.

In $\mathbb{R}^3$, the 'equivalent' would be (?) the volume of a closed object.

How is that extended to $\mathbb{R}^n$ ?

More generally, can we define such a thing in any normed space, and how do we calculate it ? (if not, in which spaces can we ?)

Part 2

In $\mathbb{R}$, the length of a segment can only diverge if the $x$ coordinate diverges towards $-\infty$ or $+\infty$, therefore there are two ways to get to infinity (?)

In $\mathbb{R}^2$, the length of a segment can diverge if the $x$ or the $y$ coordinate diverges towards $-\infty$ or $+\infty$... Therefore there are infinitely many ways to get to infinity (?).

However, if we now look at $\mathbb{R}^3$, the length of a segment can diverge if the $x$ or the $y$ or the $z$ coordinate diverges towards $-\infty$ or $+\infty$... Therefore there are infinitely many ways to get to infinity too... But this time, there are also infinitely more ways to do that than we had in $\mathbb{R}^2$ (?) !

Therefore, how can we know the infinity of ways we have to make the length of a segment diverge in a normed space $\mathbb{K}$ ? Is it $\aleph_{\dim\mathbb{K}}$ for $\dim\mathbb{K}\geq2$? (just a totally wild guess)

(be aware, even though i've used $\aleph$ in the line right before, i know about nothing about the different kind of infinities, so please post detailed answers/comments or i'll be totally lost.)

Thank you.

What you are discussing is the notion referred to as measure. While there are different notions of measure, the one that is most commonly used is that of Lebesgue measure. The $n$-dimensional volume of a set in $\mathbb{R}^n$ can be taken as the Lebesgue measure of that set.

In any Euclidean space, it is indeed as you say: a set can only have infinite measure if it is unbounded, which in a sense means that it "extends to infinity".

Those links should give you a place to start looking. I hope that helps.

Also: the cardinality of directions of motion from a particular point in $\mathbb{R}^n$ will be $2$ if $n = 1$ and $\mathfrak c^{n-1} = \mathfrak c$ (the cardinality of real numbers) for any $n > 1$.

Addendum: note that it is not always possible to define an intuitive notion of "volume" in a normed space. In particular, the space $\ell^\infty(\mathbb{R})$ of bounded real sequences is an infinite dimensional normed space in which our traditional notions of assigning sets finite volumes falls apart.

• What are directions of motion ? – Hippalectryon Jun 18 '14 at 19:08
• The question "how many directions of motion are there" is a slightly more precise rendering of "how many ways are there to get to infinity". For each "direction of motion", there is a unique ray emanating from that point "towards infinity". – Omnomnomnom Jun 18 '14 at 19:12
• How can we have $c^{n-1}=c$ with $n>1$ ? – Hippalectryon Jun 18 '14 at 19:15
• See this link for an example – Omnomnomnom Jun 18 '14 at 19:19
• To be sure, as we talk about the cardinality of "directions of motion", we're really referring to the cardinality of $n-1$ dimensional projective space. – Omnomnomnom Jun 18 '14 at 19:23